product-topology on $\mathbb{R} \times X$

Question

Let $(X,||\cdot||_X)$ be a Banach space and $|\cdot|$ the absolute value on $\mathbb{R}$. How to show that $||(t,x)||:=|t|+||x||_X$, $(t,x) \in \mathbb{R} \times X$ induces the product Topology?

Answer 1

Hint: We have $(t_n, x_n) \to (t, x)$ iff $t_n\to t$ and $x_n\to x$.